Kamarianakis-Prastacos (n.d.) Multivariate Hierarchical Bayesian Space-Time Models in Economics

Summary

Applies multivariate hierarchical Bayesian space-time modeling to Gross Regional Product (GRP) data for 51 Greek prefectures across three economic sectors (primary, secondary, tertiary) over 1980–1994. The model has four layers: a measurement error stage for observed GRP, a large-scale trend stage (linear trend with spatially varying intercepts and slopes), a small-scale spatio-temporal dynamic stage (vector autoregression, VAR, with cross-sector and cross-location dependencies), and conjugate hyperpriors. Estimation via Gibbs sampler (20,000 iterations); one-step-ahead forecasts compared to three separate Space-Time Autoregressive Moving Average with eXogenous inputs (STARMAX) models. The Bayesian model's forecasting performance is competitive with the sector-specific alternatives while capturing cross-sector and cross-prefecture interdependencies jointly.

Key Claims

• Four-stage hierarchy. Stage 0 (measurement error): $Z(i,s,t) \sim N(Y(i,s,t), \sigma^2_{i,s,t})$ — observed data = true process + measurement noise. Stage 1 (large-scale): $Y(i,s,t) = L_t(a(i,s),\beta(i,s)) + S(i,s,t) + \varepsilon(i,s,t)$, with $L_t$ a linear trend varying by sector-prefecture. Stage 2 (small-scale): $S_{t+1} = BS_t + e_{t+1}$, a VAR with spatially structured coefficient matrix $B$ and sector-interdependency terms. Stage 3: conjugate priors on all stage-2 parameters.
• Multivariate extension. Unlike the companion solo paper, the $S$ process models three sectors simultaneously, with cross-sector autoregressive terms $f_1(i,s)S(j,s,t)$ and $f_2(i,s)S(k,s,t)$ capturing inter-sectoral dynamics within each prefecture.
• Non-symmetric economic-adjacency neighborhood. Rather than geographic contiguity, neighbors are defined by economic strength: for mainland prefectures, neighbors are geographically adjacent prefectures with larger economies; for island prefectures, neighbors include mainland ports with ferry connections. Crucially, the relationship is non-symmetric (if $s_1$ is a neighbor of $s_2$ for sector $i$, $s_2$ need not be a neighbor of $s_1$). At most two neighbors per prefecture for parsimony.
• Identifiability between stages. $Y(i,s,t)$ and $S(i,s,t)$ appear jointly in the first-stage equation; $\sigma^2_Y$ and $\sigma^2_S$ are not separately identified in a classical sense. Proper Bayesian priors allow estimation, but sensitivity analysis is recommended — and conducted, showing remaining parameters are robust to variance prior changes.
• Gibbs sampler convergence. Gelman-Rubin criterion applied; convergence reached after 2,500 iterations. All Markov chain Monte Carlo (MCMC) autocorrelations drop to insignificance after first lags. Posterior means used as point estimates (neglecting burn-in).
• Forecasting comparison. Hold-out sample: 1994 GRP. Bayesian hierarchical model vs. three separate sector-specific multivariate STARMAX models. Performance competitive; model captures sector interdependencies that STARMAX models (estimated separately) cannot.
• Sectoral convergence patterns. Preliminary regression analysis shows 84% of sector-prefecture GRP trends are significantly positive (linear growth). Primary sector shows weaker spatial correlation; tertiary sector shows Athens (Attiki) dominating as the principal "neighbor" for most prefectures.

Concepts Introduced or Extended

• Bayesian Hierarchical Model
• Gibbs Sampler

Entities Mentioned

• (none with existing wiki pages)

Quotes

"Such specifications provide simple strategies for incorporating complicated space-time interactions at different stages of the model's hierarchy, and the models are feasible to implement in high dimensions."

My Take

A companion empirical paper to Kamarianakis's solo methodological paper — the two papers together illustrate the theory (solo) and an application (joint). The non-symmetric economic-adjacency neighborhood is an interesting modeling choice that reflects the economic geography of Greece more accurately than simple contiguity. The forecasting comparison is limited (one held-out year, limited alternative models) but serves as a proof-of-concept. The identifiability issue between the measurement-error and state-process variances is a genuine concern that the authors note but do not fully resolve. Both papers are undated working papers from FORTH, Greece, likely circa 2003–2005.